This one isn’t strictly about math, but it has a connection. Read on to see. Dan Meyer has posted a critique about the West Wing’s misogyny a couple times now and as he adds to it, I begin to wonder some things. Yes, the West Wing has some strong elements of misogyny in it (thanks for ruining that for me, by the way, as I was better at ignoring it before). I wonder, though, about Dan’s wondering as to how “liberals” can ignore that. While I disagree with Congressman Skinner in the show, this situation lets me understand his position a little better: he’s a gay Republican and he explains it by noting that he agrees with 95% of his party’s platform and that not everything in his life has to be about being gay. I’m pretty liberal, generally, but I don’t want that position to stop me from enjoying some (overall) high quality, intelligent writing and entertaining television.

Dan’s extended comment about the West Wing poisoning the liberal vision of government also makes me wonder some things. To me, the show tried to portray government as it could be (“with all its failings in the past, and in times to come”), as “an instrument of good”, as a “place where people can come together.” Yes, I’m using quotes from the show to illustrate my points, don’t judge me. Is this not what “liberals” want government to be? Further, regarding the idea that the show portrays the process as being “a group of smart people in a room” and the critique of the complex legislation that came out of the Obama administration, I wonder why we are demonizing complex solutions to complex problems? Is simplicity always the most virtuous option? This question is relevant in government as well as in mathematics, physics, and epidemiology, among others. Does the incessant search for a more “elegant” or “sophisticated” solution leave us disconnected from what is, by all accounts, a highly complex reality? Running a nation of 330 million people is a complex endeavor. So is describing the fundamental mechanisms and evolution of a (potentially) infinite universe over the course of 13 billion years. So is attempting to understand the spread of a novel virus in a human population of 7 billion persons. Simplicity is not our ally in those searches for understanding.
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Lastly, is it now true that in order to be an orthodox liberal, I have to allow it to permeate every portion of my life, my intellect, and my personality? Am I only allowed to watch (and want to watch) programming that portrays a world that aligns precisely with the ideals of the liberal movement? To be clear, I know the answer is no, but Dan’s post would seem to imply that my liberalism should stop me from enjoying some damned good (if flawed) television. I should not and will not relinquish my claim to liberalism because I enjoy the West Wing. While clearly it has stopped Dan from enjoying it, and I respect that, I hold no such compunction and no one else should feel that way either.

In my small Twitter sphere recently there has been a resurgence of conversation around a controversial question:

Which is more effective for students: discovery learning or explicit instruction?

I entered this conversation by reading a post by Michael Pershan, called “Discovery learning vs. not discovery learning.” Michael structures his post as a conversation between a proponent of discovery learning and a proponent of much more guided instruction. I found the post fascinating even though I completely disagreed with most of it. But it might not be for the reasons you think. Here are some issues I might advocate for us as a community to clear up.

     1. There is no such thing as “discovery learning.” 

I can’t say this loud enough or often enough. And I’m not the only one. Mathematics education is not a field that advocates for discovery learning. Further, high quality mathematics instruction is not synonymous with any of the terms used commonly in these conversations. The source of much of this equivocation seems to be a paper by Kirschner, Sweller, and Clark published in the Educational Psychologist entitled “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching.[i]” I find the arguments in this paper to be unconvincing at best. But the authors are correct about some things. Discovery learning did fail. It took the human race thousands of years to develop even the basic ideas of school mathematics, to say nothing of the full scope of modern mathematics. The idea that we can sit a student down and have them, individually, discover all of that in 13 years is ludicrous. Which is why we don’t do that.
 
So here we are. High quality mathematics instruction—that’s what I’ll call it from now on—is not discovery learning. It is not, strictly speaking, constructivist. Nor is it in any way “minimally guided.” The remaining three names have connections to high quality mathematics instruction, to be sure. However, it is more than any one or two (or even all three) combined. High quality—can I just call it HQMI, for short?—HQMI uses problematic situations to create mathematical experiences for students that involve making sense of those situations and using that to develop or choose a solution pathway or to consider an existing method. These experiences are highly social in nature, involving both individual and collaborative work and sense-making.
 
The role of the teacher is extremely important in every phase: planning, launch, explore, summary discussion, assessment. The teacher is pivotal in launching the task so that students’ prior knowledge is activated and the class understands the context well enough to begin work. The teacher structures the middle portion of the lesson so that students have both individual and collaborative thinking time, supporting them with questions that uncover and focus students’ thinking. For the final phase, the teacher decides the order of presented and discussed solution strategies. Here the teacher’s role is to clarify students’ understandings, engage them in comparing methods, and drawing connections between those methods (e.g., which might be most efficient, which is most “elegant,” etc.). Assessment isn’t a phase as it happens throughout the lesson, with the goal being to learn about how students are thinking about the mathematics.
 
None of this. Absolutely none of it, would happen without the presence and active engagement of the teacher. The goal is not for each student to “discover” everything. The goal is for the class to engage, as a group, in sense-making and to construct a group understanding of important mathematics.
 
     2. What the mathematics education community, generally, values is sense-making. 

I’ve already hinted at this one, but it deserves some attention on its own. Michael is right in his post when he says that “[t]here comes a point where people just disagree on what they value. It’s hard to know what to say by the time someone gets to this point of clarity about what they care about.” James Hiebert[ii] makes this point very well when he says that “[d]ebates about what the research says will not settle the issue; only debates about values and priorities will be decisive. Until the value issue is settled, it will be difficult to find common ground for examining the research.” (p. 5).
 
So let’s talk about values. Let’s talk about priorities. The mathematics education community has been steadfast in its valuing of sense-making for more than 30 years at this point. Sense-making and flexible thinking are given priority in every document published by NCTM[iii], NCSM[iv], AMTE and many of the other professional mathematics organizations. Sense-making is given priority in the design of the Common Core State Standards for Mathematics, the Standards for Mathematical Practice, and nearly any derivative document (including a majority of derivative state standards). In short, the math wars are over. Sense-making is valued in our community.
 
This informs our conversation moving forward. Hiebert again helps us out: “We now know that we can design curriculum and pedagogy to help students meet the ambitious learning goals outlined in the NCTM Standards. The question is whether we value these goals enough to invest in opportunities for teachers to learn to teach in the ways they require.” (p. 16).
 
     3. There is a body of research in mathematics education that supports HQMI as more effective at producing student                learning than direct instruction. 

Given the answer to the last question, there is some research worth examining. First, most research that compares approaches characterizes direct and/or explicit instruction as “traditional.” This will be important when you read the pieces below. One of my favorite pieces of research is Hiebert and Wearne (1993)[v]. This study, in short, studied traditional classrooms (teacher-centered, direct instruction) as compared to alternative classrooms—students “received fewer problems and spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, talked more using longer responses, and showed higher levels of performance or gained more by the end of the year on most types of items.” (p. 393). The quasi-experimental design of this study offers robust support for the results.
 
Another example would be the work of Jo Boaler and Megan Staples in the Railside School study[vi]. This study found that a poor, highly diverse school could outperform more affluent schools through the use of accessbile curriculum and ambitious mathematics teaching practices. In short, a fundamental commitment to the use of problematic situations and the engagement of students in ways consistent with HQMI can produce profound results. Not only did performance increase, but also students’ mathematical identities and relationships with mathematics were starkly different[vii].
 
Then there are the curriculum studies. NSF-funded curriculum materials at all levels have been researched extensively[viii]. Each of these curricula use instructional models that are consistent with what I’ve described as HQMI[ix]. The sum of the research in this area is as follows: students who learn from curriculum materials that are reform-oriented (i.e., like the NSF-funded curriculum projects) generally outperform students who learn from traditional models along two axes. First, reform approaches tend to produce superior (or at least comparable) performance on standardized assessments. Second, these kinds of learning experiences also tend to produce better results on measures of problem-solving ability and nearly all affective factors associated with mathematics learning.
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At this point, I’ve talked enough for one blog post. These arguments are based on my own experiences and my own understanding of the research I’ve read. I do not claim that this is a comprehensive case, but I believe that I’ve addressed many of the concerns brought up in Michael’s blog post. I have some other thoughts that I’ll get down in a sequel to this post:

1. Direct instruction prevents access to mathematical content for some students
2. Consider an example.

But, for now, I think—hope—that’s enough to keep our mutual conversation moving.


[i] Kirschner, P., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of construtivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75-86.

[ii] Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for research in mathematics education, 30(1), 3-19.

[iii] See, e.g., NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM; NCTM. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM and any associated books.

[iv] See, e.g., NCSM. (2020). NCSM Essential Actions: Framework for Mathematics Education Leadership. NCSM; NCSM. (2019). NCSM Essential Actions: Coaching in Mathematics Education, NCSM; NCSM. (2019). NCSM Essential Actions: Instructional Leadership in Mathematics Education. NCSM;

[v] Hiebert, J. & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393-425.

[vi] Boaler, J. & Staples, M. (2008). Creating equitable futures through an equitable teaching approach: The case of railside school. Teachers College Record, 110(3), 608-645. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41-62.

[vii] Boaler, J. & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adult’s lives. Journal for Research in Mathematics Education, 48(1), 78-105; Boaler, J. & Greeno, J. G. (2000). Identity, Agency, and Knowing in Mathematics Worlds. In J. Boaler (Ed.) Multiple Perspectives on Mathematics Teaching and Learning (pp. 171-200). Westport, CT: Ablex Publishing;

[viii] See, for example: Huntley, M. A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey, J. T. (2000). Effects of Standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand. Journal for Research in Mathematics Education, 31(3), 328-361; Grouws, D. A., Tarr, J. E., Chavez, O., Sears, R., Soria, V. M., & Taylan, R. D. (2013). Curriculum and implementation effects on high school students’ mathematics learning from curricula representing subject-specific and integrated content organizations. Journal for Research in Mathematics Education, 44(2), 416-463; Tarr, J. E., Grouws, D. A., Chavez, O., & Soria, V. M. (2013). The effects of content organization and curriculum implementation on students’ mathematics learning in second-year high school courses. Journal for Research in Mathematics Education, 44(4), 683-729;

[ix] Examples include Everyday Mathematics (K-5), Connected Mathematics Project (6-8), Core-Plus Mathemetics Project (9-12). For a complete list, see https://www.nsf.gov/pubs/2002/nsf02084/chap1_4.htm.

​Recently, I have seen administrators across Michigan flooding Twitter with celebratory tweets about the statewide switch to the PSAT 8 as the high-stakes accountability test for 8th grade students and teachers. Most of these tweets sound very similar: The PSAT 8 is the test we need. It provides schools, parents, and teachers with the data they need to improve. Putting aside the eerie similarity of the wording, let’s be clear about something: they are all very wrong. We need to be honest about this. We need to have an honest conversation as a state about how these ideas are misguided at best, dangerous at worst.

To begin that honest conversation, I’ll start with my contention that the PSAT 8, in fact, does none of the things these administrators and organizations claim it does. I believe this is true for at least four reasons:

1. The PSAT 8 has a fundamental purpose. It is meant to separate students.

The SAT Suite of Assessments (designed by the College Board) is designed to track one thing, and one thing only: college readiness. These tests are meant to separate students into two groups: college ready and . . . not. Here is where we hit our first snag. The policy of using the PSAT 8 for accountability purposes is in direct contention with the purpose of the assessment. The PSAT 8 is designed to be predictive of student performance on the SAT which is designed to give an indication of a given student’s likelihood of succeeding in entry level college coursework. Success is, in this case, defined as a probability of receiving a B in a credit-bearing course during freshman year.

There are a number of issues at play here. First, grade 8 educators have a responsibility to prepare students well in mathematics as defined by the 8th grade mathematics content standards in Michigan. That, in no uncertain terms, is their mission. Implementing PSAT 8 will drive schools to prepare students to perform on the PSAT 8, not on the end of year learning targets laid out in the standards which have been adopted by our state Board of Education. Second, it is always dangerous to use an assessment for a purpose other than its designed and intended purpose. This assessment is meant to measure college readiness as defined by the College Board, not students’ thinking and understanding of given mathematical content. Third, this separation and labeling of students does nothing to help those who are deemed “not college ready” by this assessment. Indeed, it likely dooms them to remedial tracks throughout high school, resulting in lowered expectations for them across the board. And this is the real danger. The assessment contributes to the gaps in student performance and learning that we currently see.
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2. The PSAT 8 is a norm-referenced assessment.  

This concern is a fundamental one in many ways. Our education system is based, currently, on standards or criteria to be met by the end of each successive school year. In such a system, assessment is best designed to determine student understanding in those areas and students’ abilities to meet the criteria laid out for them. The PSAT 8 is not designed in this way. This assessment provides an overall score in mathematics as well as several sub-section scores and oftentimes gives students information about the percentiles in which they reside. This means that the data are compared across the national cohort of students taking the test at that time. And scores are assigned relevance based on the performance of other students, not on any static criteria.
 
Normative data used for high-stakes decisions within a criterion-referenced system is nonsensical. Educators’ time is better spent considering how students understandings match the criteria we wish them to meet than considering how students compare to the average performance or to other students in their peer group. Certainly, there are “alignment documents” that give indications of which of the standards fit into the buckets of content of the assessment—but that is not the same as using an assessment designed to assess the criteria themselves.

3. The PSAT 8 provides data that is highly uninformative about classroom instruction.

One of the most detailed reports the PSAT 8 portal will provide is the Question Analysis Report. Educators love this report. They. Love. It. And for all the wrong reasons. I get it. The report gives you the performance of your students, the state’s students, and the nation’s students on a given assessment item. It shows you the percentages of students who chose each distractor. It even gives you the item to look at! The report feels like a gold mine. But ultimately, it’s fool’s gold. Here’s why.

As educators pour over this report they feel as though they are getting very accurate data about student performance. But these item-based analyses are dangerous because they easily lead to solutions designed to fix problems that may not exist. Let me illustrate. I sat with two district administrators and looked over a set of PSAT 8 data. We spent almost two hours looking at students’ performance on questions limited to the Heart of Algebra strand on the PSAT. After all of this work and my attempts to get them to see connections among the items and to come up with alternative explanations for students’ choices, the big takeaway for them was that they needed to work on systems of linear equations more.

There are two concerns that surface in the example above. First is the lack of information about students’ thinking. We certainly know which questions many students got wrong and we even know which distractors students chose most often. But any attempt to figure out why students chose those distractors is stymied by our lack of information. While it is tempting to say that the distractors were designed to take advantage of common misconceptions, that explanation is ultimately self-defeating. There are a number of potential reasons for a student to choose a given distractor, only one of which is that their misconception matches the one intended by the item writers. Without looking at student work in detail or talking to the students themselves, educators can never be certain they have even a vague idea of students’ issues. Second, when summarizing the efforts at the end of a meeting, educators are (understandably) drawn to particular examples of problems that were a struggle for students (like the systems problems in our example). But these particulars are, in all likelihood, small percentages of the kinds of problems students will likely see on a given form of the PSAT. To be clear, in the example, there were at most four items that were related to students’ understandings of systems of linear equations, a small portion of the overall assessment. This kind of item-based decision-making is dangerous and will oftentimes lead to solutions that miss the mark.

4. The PSAT 8 assesses content that grade 8 teachers and students are not responsible for.

There are two concerns associated with this issue: fairness and accountability. I would argue that it is inherently unfair to give an assessment to students which contains content that they have not learned yet, regardless of whether or not success on that content would only increase their score above the proficient mark. The PSAT 8—indeed any high-stakes assessment of this nature—contains items on which it is expected that students will not be able to perform. After all, the test has to separate students somehow, right? Here is the heart of the unfairness. The PSAT 8 contains content that is not present in the grade 8 standards because it is meant to predict performance on the SAT, not assess 8th grade content.
As for accountability, if I were an 8th grade teacher, I would be absolutely irate that I was being measured for accountability and evaluation by a tool that contained content for which I was not responsible. And because educators take their evaluations seriously, many will feel as though they have no choice but to teach the more advanced content to everyone in the hopes that it will better their evaluation results. Systemically, this might lead to a practice of requiring all 8th grade students to take Algebra 1, a practice that we learned from California is detrimental to students. And maybe that is the most insidious problem of all: with the best of intentions, these decisions are made based on the needs of adults and not on the needs of students. Accountability. School data. Allow schools to improve. Very few of these sentiments mention students specifically. And that’s because the assessment is not designed to be friendly to students—it is designed to separate and label them.

To close out this discussion and make it, perhaps, more productive. I ask the following questions:

1. What kinds of data do teachers actually need to improve their classroom practice on a daily basis?
2. What kinds of data can schools collect easily that give insight into the effectiveness of their curriculum?
3. What alternative types of assessments might allow us to get at what students know and can do?
4. How can we have an honest conversation about how we might effectively use PSAT 8 data to help schools and teachers?
   1. What can the PSAT do for us, based on its intended purpose?
   2. What can’t it do for us, based on its intended purpose?

Notice the difference?  Concept appears so much earlier in the sequence when technology is present, not to mention the fact that it is so much more efficient to use technology to produce representations.  This allows students to see many more examples in the timeframe of perhaps one or two examples in settings without technology. In the presence of technology (effectively used) the idea of representational fluency (Kieran, 2007), always central in algebra courses, now becomes much more accessible to many more students.  Here I’m reminded of a quote on Twitter attributed to Jo Boaler that goes something like “If there are more ways to be successful, then more students will be successful.”

Technology allows and facilitates student success through both efficiency in creating multiple representations and through its ability to display dynamic, multimedia content. Videos, blogs, songs, clickable, drag-able entities, animations, programmable applications—the options are almost limitless.  And none require that kids sit in rows of cubicles.  However, these things do not come naturally to either teachers or students and so another thing is needed.

Thought 2: Any technology infused into a classroom should come with both a plan and the appropriate training for all stakeholders.

Remember my Field of Dreams reference? Well here is the remedy.  Plan. It’s as simple as that. But I don’t mean plan as in “Run the numbers. Can we afford this? How many will we need?” No, I mean plan as in “What do we want technology to do for us? What does research say is the most effective use of technology in the classroom? Who should have it? How much should they have? Should we phase in or flood? How will we ensure that this tech is used? What training is needed? What lessons have other schools learned?” These are just a few of the questions, in addition to the cost questions, that should be answered thoroughly in any instructional technology initiative.

For mathematics education in particular, NCSM (National Council of Supervisors of Mathematics) has authored a white-paper with CoSN (Consortium for School Networking): Mathematics Education in the Digital Age.  This paper would make a wonderful starting point for talking about the questions I asked above.

Another interesting thought about planning for technology implementation is the idea that personal use and classroom use are two very different things for both teachers and students. This idea was brought home to me by a presentation at a CSMC conference in which a teacher talked about how the district had carefully planned for the introduction of iPads into their district.  They made the devices available to the teachers for a year before they gave them to students in the hope that teachers would have time to “get used” to the new tech before students got it. This seemed like a good plan until the next year when school officials realized that some teachers (but not all) had gotten used to the technology, but for using it in their personal lives not their classroom instruction.  Teachers could check their emails and could go to websites, but really had no idea how to use the technology to instruct. This disconnect, for them, underscored the need to plan better.  To take a different view of technology implementation.

A Final Thought: Haste makes waste.

Yes, I know I just resorted to cliché.  However, in this case it is the literal truth. When it comes to technology in the classroom, nothing is a greater waste than to have staff and students doing nothing but checking email and surfing social media on the new technology provided by the school. 

So don’t be in a hurry for the mechanized, pixilated vision from the beginning of this entry. Not only is it too early for such a thing to truly be possible, but I don’t believe that it will ever be preferable. Beware of that oft-touted holy grail: Personalized Education—it’s a dangerous myth. If you are going to implement technology in your classroom or in your district, make sure you plan first.  Plan well.  Plan much. And above all, ask questions.  Know the research, ask the experts, ask those who have gone before you.

Technology has great potential, but like any tool it must be used wisely. How wisely are you prepared to use it? 

References

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics      education. Journal for Research in Mathematics education, 2-33.

Kieran, C. (2007). Learning and Teaching Algebra. Second handbook of research on mathematics teaching and learning:        A project of the National Council of Teachers of Mathematics, 1, 707.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge university press.

Recently, I sat through an "education town hall meeting" sponsored by a conservative political candidate in my state region.  Normally, I do not attend such things as my interest in politics has waned severely in recent years.  However, this candidate caught my interest because of the prominently displayed "Anti-Common Core" statements in the advertisements for the event.  Being very much pro-Common Core myself, I decided to do a little bit of opposition research.  I would just go and hear what the candidate had to say. 

Well, needless to say, things did not go exactly as I had envisioned.  The first sign was the presence of the leader of the state Anti-Common Core movement.  This person took over the presentation almost immediately and then my problems began.  The presentation was a well organized, well constructed, and well delivered.  Unfortunately, it was also a completely unfounded and patently untrue critique of the Common Core State Standards.  By the end of the presentation I was extremely upset and agitated, which is unusual for me.  My objection rose not from the fact of their opposition to the standards, but rather from the polished, assured delivery of complete untruths.  Now, I am not accusing anyone of lying.  The presenter and candidate might even believe what was said, but the fact of the matter is that nearly all of the information given was incorrect, misinterpreted, or untrue.  I left the meeting very disturbed by the thought that this information is being given to many people in my state and that (even presuming positive intentions) it might have a drastically negative effect on legislation and community perception of education.

Later that week as I began my summer reading list, I came across a quote attributed to John F. Kennedy when he spoke at Yale University in 1962.

"As every past generation has had to disenthrall itself from an inheritance of truisms and stereotypes, so in our own time we must move on from the reassuring repetition of stale phrases to a new, difficult, but essential confrontation with reality. For the great enemy of truth is very often not the lie-- deliberate, contrived, and dishonest-- but the myth-- persistent, persuasive, and unrealistic . . . Mythology distracts us everywhere."

This quote struck me immediately and has consumed my thoughts since.  Coupled with my experience at the education town hall meeting, it started me thinking about the nature of the Common Core Mythology (I choose the term myself).  The growth and organization of the opposition movement over the past several years has brought with it a persistent, pervasive, and to some people, persuasive mythos (Thank you, President Kennedy, for those wonderful words).  This mythology is built upon the fears and tacit acceptance of those who believe public education to be fundamentally flawed.  It is, at its core, a misinformation campaign that relies on the fears and apathy of its victims to thrive.  To those with political agendas it is a call to arms to gather support from parents afraid that their children are being destroyed or brainwashed, from disillusioned educational professionals (not just teachers, but anyone in the educational family), and from those who adamantly oppose change in any form.  I believe that those of us who understand the standards and are not afraid of them have a duty to the truth about these standards and our children.  All of this has led me to ask several questions.

                1. What is the "Common Core Mythology?"

                2. What stereotypes and truisms (both true and false) have arisen with these standards?

                3. What are the reassuring stale phrases of the Common Core Mythology?

                4. How can we come to our "essential confrontation with reality" about education and the Common Core              Mythology?

The first two questions are best answered together.  Because the Common Core Mythology is really all of the misinformation, lies (well, half-truths at the very least), and misunderstandings surrounding the standards.  It is the rehashing of old feuds and stereotypes from other eras of education reform.  Here are some statements I've heard consistently.  All of them are, at least in part, untrue.

                "Common Core is a national curriculum."

                "Common Core is federally mandated."

                "Common Core tells teachers what to do every day and doesn't let them be professionals."

                "Common Core mandates tests that are too hard for our students."

                "Common Core is brainwashing our kids with "fuzzy math," liberal ideals, and morally objectionable material."

                "Common Core requires homework that I can't even do, and I have an advanced math degree."

                "Multiplication proficiency is defined differently."

Whatever the turn of phrase opponents use, it always seems to be based in one side of a dichotomy that most people don't think about: intent versus implementation.  I've seen many different examples come through social media (particularly the "I can't help my son/daughter with  Common Core homework."), and while I don't doubt that the assignments and issues being discussed are very real and concrete, they come from teachers who have the best of intentions.  Maybe this is an integral part of the Common Core Mythology as well.  That the standards will be implemented with equal effectiveness in all schools.  This is obviously untrue, as any set of standards is subject to interpretation and any implementation plan can go awry because of the scale involved.  I am not saying that these  things are okay.  I am saying that they can and will happen, and perhaps everyone should presume positive intent before casting harsh judgments.  By "presume positive intent," I mean that teachers charged with implementing these standards in schools do not get up in the morning planning on destroying the lives and education of children.  That should be taken into account before casting aspersions on someone's work.  Perhaps a civil, reasonable conversation should be used in lieu of a stream of vitriol and bile spewed up all over Facebook and Twitter feeds.

The role of the federal government is most certainly part of the mythos, as evidenced by the first few statements above.  This is the area at the heart of most opposition positions.  However, I must be very clear on this point: these standards were purely voluntary.  With that being said, yes the federal government did step in and incentivize the process with Race to the Top funding.  While some may disagree about the appropriateness of this move, I do not believe any can deny the extent to which it enhanced the effectiveness of the Common Core State Standards Initiative.  Within a few weeks nearly every state in the union had signed on to the project.  Even the most notable exception, Texas, when viewed closely, has a set of standards very much like the Common Core.  Indiana, after backing out very recently, adopted a very similar position in their new standards.  Even the Fordham Institute rated these standards quite highly in both Mathematics and English Language Arts.

As I read this piece over once more, I realize that I, sadly, do not have an answer to my fourth and final question.  But perhaps that's just as well.  Perhaps this discussion is the beginning of that confrontation with reality.  There is much more to say.  Many more debates to be had in this fertile arena.  From here I send the discussion your way as I look forward to your thoughts concerning the Common Core Mythology.  I've begun to describe the rise and attempted to begin the fall.  Where do you stand?

How would you teach students to divide fractions?  As in, what would you do on Day 1 of your dividing fractions unit?  I've asked this question of teachers many times in professional learning settings and all too often the answer I get is . . . wait for it . . . "I'd teach them to flip the second and multiply."  Those of us who learned fraction division at some point in the last, say 50 years, are very familiar with this "standard" algorithm for dividing two fractions. 

Fraction division is not the only operation that has a "standard" algorithm, of course.  Almost all operations in math have one.  Most, if not all, of us are familiar with the Fearful Four of standard algorithms:

·         the "carrying/borrowing" algorithm for addition and subtraction of whole numbers and decimals,

·         a similar algorithm for multiplication of whole numbers and decimals,

·         Long Division (yes, capitalized) for division of whole numbers and decimals,  

·         and the dreaded "invert and multiply" for division of fractions. 

Yes, mathematics has hundreds more smaller, less well-known standard algorithms (reduction of fractions, converting between mixed numbers and improper fractions, etc.) but the four above are the biggies.  They also happen to be the four that are mentioned explicitly in the K-8 Common Core standards. 

Well, almost explicitly.

The "carrying/borrowing" algorithm is often referred to as the "standard" algorithm for addition and subtraction, but the standards do not specify that particular algorithm.  They simply say "Fluently add and subtract multi-digit whole numbers using the standard algorithm." So there has been some debate over the definition of standard algorithm, and it turns out that this definition is not universal.  Some experts and math education leaders (and I agree with them) would suggest that the "standard" algorithm for addition and subtraction should be partial sums.  The standard for multiplication could be partial products.  Both of these methods have advantages over the traditional "carry/borrow" algorithm in both connection to place value concepts and encouraging flexibility in student thinking. 

It is also worth noting that "Fluently add and subtract multi-digit whole numbers using the standard algorithm." is standard 4.NBT.B.4 (CCSSI, 2010).  This means that the standard algorithm is pushed off until 4th grade, after students have learned to add and subtract multi-digit whole numbers in Grades 2 and 3, with introductions from early Kindergarten through Grade 1.  The algorithm is pushed off until very late in students' experiences with those operations. 

Much like the punch line to a joke. 

While the punch line metaphor is perhaps unflattering, it is nevertheless effective and accurate.  Students build understanding of operations through experiences and informal reasoning, just as listeners build understanding of the amusing situation in the body of a joke, and only after having sufficient experience with and understanding of the operations are they ready to learn the algorithm with understanding, just like the listeners and the punch line.

Another aspect of this debate is whether students should even be taught certain algorithms in school.  David Ginsburg blogged about the necessity of long division, so I'll avoid that one here (but you should all read So Long, Long Division) and move on to some others.  I've no argument against the standard algorithms for most operations in elementary schools but I'll choose one that I do have an argument against: reduction of fractions (or writing fractions in simplest form for all my high school teachers out there).  This is the "punch line to bad jokes" portion of the blog.

And don't worry, I'll get to a hot button secondary example in my next post.

But first, reduction of fractions.  Before everyone piles on all at once here, I am absolutely not saying that students should not be taught how to write fractions in simplest form.  What I am saying is that we should not teach it as an algorithm and we should definitely not require students to write every single fractional answer they ever get in simplest form.  This is because in many mathematical situations, the reduced form of a fraction will not reveal enough information about the initial problem context.  Not to mention all of my statistics friends out there who will argue tooth and nail that a probability of three-fourths is absolutely not the same as a probability of six-eighths (and they are right).  That being said, I do acknowledge the necessity of reducing fractions and so I propose teaching the idea or procedure with understanding.

My reasoning for this is two-fold.  First, we spend a great deal of time in schools beating students over the head with reducing fractions and we never tell them why or require that they make sense of the actual procedure.  Time that could be better spent both exploring more important topics and showing students success instead of highlighting failure.  Distributed practice is very important, however when students are completely correct in reasoning and even calculation on a problem but receive bad marks due to an answer that is unreduced, teachers are doing them a great disservice.  We must ask ourselves what we are trying to assess about students' knowledge and be sure we aren't needlessly penalizing them for something that is quite trivial and unrelated to the knowledge base we desire.

Second, students will be much more likely to remember how to "reduce" fractions if we teach them under the guise of generating equivalent fractions.  The CCSSM focus on equivalent fractions in Grades 3-7 should be continued throughout all grades.  Students are very capable of generating equivalent fractions from almost the beginning of their study of fractions and we should nurture this instinctual understanding, not drive it out with a meaningless procedure.  Students can understand equivalent fractions in  a few different ways, all of which are very effective and still allow students to find the "simplest" form of a fraction.  One great example I've heard before is that of a reduced fraction being the "simplest recipe" in a given situation.   This idea, in conjunction with appropriate contexts, can really drive home the idea of reduced fractions with understanding.

So the moral of the story today is: teaching standard algorithms is fine if you must.  Just treat them like the punch line of a joke and don't deliver them until the end of the learning trajectory.  Remember, procedural fluency should be developed from conceptual understanding, not the other way around.

Those are some of my thoughts.  I'd love to hear your thoughts on the subject!

So, I need some help here.  This is my first blog post and I though I'd use it to get some feedback on a project I'm working on!  I am the Director of the NCSM Fall Leadership Seminar Series.  Well, co-director, anyway.  NCSM is an organization dedicated to leadership in mathematics education and the Fall Leadership Seminars are single-day professional learning opportunities focused on leadership topics, resources, or any number of other things the directors might decide to include. Here are some of my thoughts on the content of the seminars.  

In planning for the 2014 NCSM Fall Leadership Seminar Series, I looked back over the 4 years since the release of the Common Core State Standards for Mathematics trying to take stock of where we’ve been and where we are now in our path toward effective implementation.  I took note of the struggles many states have seen and, of course, all of the truly prodigious efforts of mathematics teachers and mathematics education leaders everywhere.  We have made truly impressive progress and we are capitalizing on many of the opportunities that this movement provides.  We do, however, still have very far to go in this country to move mathematics education into something resembling our desired state.

So this year, after much consideration and discussion, the Directors have decided to use the Fall Leadership Seminars as a time to look ahead to the future of mathematics education.  We are nearing the midpoint of the second decade of the 21st century and never has it been more apparent that this time affords us ever more unprecedented opportunities in math education based on technology.  The advances continue apace to those of the past ten years and even seem to be accelerating.  The lists of apps, and programs, and websites devoted to various aspects of mathematics education are endless and growing faster every day.  These lists are difficult, if not impossible, to keep up with!  As leaders in mathematics education it is our responsibility to help ourselves and our teachers sift through this deluge of material, finding the most promising and the most effective.  But how is this done?  Are there criteria useful in creating this modern day Sieve of Eratosthenes?  How should a truly digital learning environment look?  What are the best practices?  The nature of digital textbooks is changing as well.  How should districts choose these resources?  What are the hallmarks of a quality digital textbook?  These questions are difficult and require the input of experts and practitioners everywhere.  The Fall Leadership Seminars will address some of these questions and help leaders develop a vision for the future of technology in mathematics education. 

While technology is certainly important in today’s learning environments, it is not the only area of vital concern.  As we push toward the higher standards of the CCSSM, we must utilize every tool at our disposal and take advantage of the many opportunities that this era in mathematics education provides.  I've often heard NCSM President Valerie Mills speak of formative assessment and its power in the classroom and I’ve found myself in agreement time and again.  Formative assessment is one of the most powerful tools available to us as a means of increasing student learning.  Research has borne this statement out many times.  Those familiar with the work of John Hattie in Visible Learning have read of the effectiveness of this classroom practice.  We would be remiss if we did not avail ourselves of a tool of such power.  Yet, despite all of that, formative assessment remains a practice that is often overlooked and little understood.  The Fall Leadership Seminars in 2014 will examine both the potential role of technology as well as effective formative assessment practices.

The Fall Leadership Seminars will meld these two themes this year.  We will explore the future of the mathematics classroom through the interplay of technology and formative assessment. We will be meeting on three days in three cities this fall:

Wednesday, October 29, 2014 in Indianapolis, Indiana

Wednesday, November 12, 2014 in Richmond, Virginia

Wednesday, November 19, 2014 in Houston, Texas

Thank you to all of you that attended the seminars last year!  You made the series a great success!  Denise and I invite all of you to join us at this year’s NCSM Fall Leadership Seminars!  

Thoughts?  Ideas?  This is relatively uncharted territory for everyone and so I welcome any and all input!  I want to put together something that mathematics education leaders will want to see!